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Lecture Summaries for Differential Equations

Lecture 01: An introduction to the very basic definitions and terminology of differential equations,
as well as a discussion of central issues and objectives for the course.

Lecture 02: Solving first order linear differential equations and initial value problems using
integrating factors .

Lecture 03: Solving separable equations.

Lecture 04: The Existence and Uniqueness Theorem for solving general first order linear equations.

Lecture 05: Applications of first order ODEs involving continuous compounding, and population
dynamics using the logistic equation .

Lecture 06: Solving the logistic equation, and an application of first order ODEs to a problem
of physics.

Lecture 07: Solving exact equations.

Lecture 08: Sketching a proof of the Existence and Uniqueness Theorem for first order ODEs.

Lecture 09: An introduction to difference equations and their solutions, focusing on first order
linear difference equations.

Lecture 10: An application of first order linear difference equations, as well as a brief discussion
of non- linear difference equations, their solutions, and stairstep diagrams.

Lecture 11: An introduction to second order ODEs and initial value problems, and a discussion
of solutions to second order homogeneous constant coefficient equations.

Lecture 12: A discussion of existence and uniqueness results for second order linear ODEs, and
of fundamental sets of solutions and the importance of the Wronskian of solutions.

Lecture 13: A discussion of the structure of the set of solutions to a linear homogeneous ODE
from a linear algebra perspective ; concepts such as linear independence, span, and basis are used
to better understand fundamental sets of solutions .

Lecture 14: Solving ODEs with characteristic equation having non-real complex roots.

Lecture 15: Solving ODEs with characteristic equation having repeated roots.

Lecture 16: Solving second order linear non-homogeneous equations using the method of undetermined
coefficients .

Lecture 17: Solving second order linear non-homogeneous equations using the method of variation
of parameters.

Lecture 18: A discussion of the structure of solution sets to higher order linear equations, the
basic Existence and Uniqueness Theorem, and a generalization of the Wronskian.

Lecture 19: Solving higher order constant coefficient homogeneous equations.

Lecture 20: Solving higher order non-homogeneous equations using the method of undetermined
coefficients .

Lecture 21: Solving higher order non-homogeneous equations using the method of variation of
parameters.

Lecture 22: A review of the most fundamental properties of power series.

Lecture 23: Solving differential equations and initial value problems using power series.

Lecture 24: An example of how to use power series to solve non-constant coefficient ODEs,
and a discussion of the basic theorem underlying the use of power series to solve ODEs.

Lecture 25:
A review of improper integration and an introduction to the Laplace transform.

Lecture 26:
A discussion of the main properties of the Laplace transform which make it useful
for solving initial value problems.

Lecture 27: A discussion of how the Laplace transform and its inverse act on unit step functions,
exponentials , and products of these functions with others.

Lecture 28: An introduction to the convolution of two functions, and an examination of how
the Laplace transform acts on such a convolution.

Lecture 29: An introduction to systems of equations and the basic existence and uniqueness
result for the corresponding initial value problems .

Lecture 30: An introduction to vector function notation, and a discussion of the structure of
solution sets to homogeneous systems and the importance of the Wronskian.

Lecture 31: Solving constant coefficient linear homogeneous systems using eigenvalues and
eigenvectors.

Lecture 32: Solving constant coefficient linear homogeneous systems in the case where an
eigenvalue is complex .

Lecture 33: Solving constant coefficient linear homogeneous systems in the case where there is
a repeated eigenvalue.

Lecture 34: Viewing solutions to linear homogeneous systems in terms of fundamental matrices
and the exponential of a matrix .

Lecture 35: Solving non-homogeneous systems using diagonalization and variation of parameters.

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